edit: It's also interesting to note what happens if, say, your potential energy is not Hermitian. I recommend you explore that exercise a bit, as you get some interesting results regarding the normalization of the wave function. Also, it's important to note that most non-conservative forces (such as friction) are macroscopic, and, as far as I know, have no quantum analogs.

The S.E. in 'fundamental form' is [tex]i\hbar \frac{\partial}{\partial t}|\Psi\rangle = H|\Psi\rangle[/tex].
You should always get H from the classical Hamiltonian. The S.E. you wrote down cannot accomodate for all situations.